What Is the Rule of 72?
Years to double are roughly 72 divided by the yearly rate in percent.
Free lesson
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In this lesson you learn to
- Estimate doubling time with the rule of 72
- Work backwards: estimate the rate needed to double in a given time
- Know where the approximation is close and where it drifts
Check yourself
Think about each question, pick what you would answer if you like, then select “Show answer” to see the solution and why. Nothing is graded or sent to us. Example numbers are worked examples, not predictions.
1.What does the rule of 72 say?
Years to double are roughly 72 divided by the yearly rate in percent.
2.At 9 percent a year, about how many years until money doubles by the rule?
72 divided by 9 is 8 years, and the exact answer is just over 8.
3.At 6 percent the rule gives 12 years. What is the exact answer, to one decimal?
The rule gives 12 years; the exact calculation gives 11.9 years.
4.Leo earns 8 percent a year. How many years per doubling does the rule give?
72 divided by 8 is 9 years per doubling. For Marge at 4 percent, 72 divided by 4 is 18 years.
5.Marge and Leo each start with 1,000. After 36 years, Leo at 8 percent has doubled four times. About how much does he have?
Leo doubles four times: 2,000, 4,000, 8,000, about 16,000. Marge doubles twice, to about 4,000. Twice the rate gave Leo four times the money, not twice.
6.Running the rule backwards: to double in 6 years, roughly what yearly rate is needed?
72 divided by 6 says it takes about 12 percent a year.
7.If prices rise 3 percent a year, about how many years until they double, by the rule?
72 / 3 = 24 years; the exact value is 23.45. The rule works for anything that grows by a steady percentage.
8.Which of these numbers divide 72 exactly? Select all that apply.
Several answers can be correct.
72 is divisible by 2, 3, 4, 6, 8, 9 and 12, so most everyday rates give a clean answer.
9.True or false: the rule of 72 is exact at every interest rate.
The rule is close between about 4 and 12 percent, and drifts outside that range.
10.At 1 percent a year the rule says 72 years. About what is the true doubling time?
At 1 percent the rule says 72 years, but the truth is about 70.
11.At 36 percent a year the rule says 2 years. What is the true doubling time?
The higher the rate, the further the shortcut drifts: 2 years by the rule, 2.25 years exactly.
12.True or false: for continuous growth, specialists use 69.3 instead of 72.
For continuous growth, doubling time is ln 2 divided by the rate, which gives 69.3 divided by the rate in percent.
Sources (2)
- MacTutor (University of St Andrews): A Napierian logarithm before Napier (Pacioli, Summa de arithmetica 1494, fol. 181, n. 44)
- Eigene Berechnung: exakte Verdopplungszeit t = ln 2 / ln(1 + r), stetig t = ln 2 / r; 100 * ln 2 = 69,31
Education only. Not financial advice.